Wigner’s quasi-probability distribution function in phase-space is a special (Weyl–Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics, nuclear physics, quantum computing, decoherence, and chaos. It is also of importance in signal processing, and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory: The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways.

Here are some quotes. First, the phase-space formulation should be placed on equal footing with the Hilbert-space and path-integral formulations:

When Feynman first unlocked the secrets of the path integral formalism and presented them to the world, he was publicly rebuked: “It was obvious”, Bohr said, “that such trajectories violated the uncertainty principle”.

However, in this case, Bohr was wrong. Today path integrals are universally recognized and widely used as an alternative framework to describe quantum behavior, equivalent to although conceptually distinct from the usual Hilbert space framework, and therefore completely in accord with Heisenberg’s uncertainty principle…

Similarly, many physicists hold the conviction that classical-valued position and momentum variables should not be simultaneously employed in any meaningful formula expressing quantum behavior, simply because this would also seem to violate the uncertainty principle…However, they too are wrong. Quantum mechanics (QM) can be consistently and autonomously formulated in phase space, with c-number position and momentum variables simultaneously placed on an equal footing, in a way that fully respects Heisenberg’s principle. This other quantum framework is equivalent to both the Hilbert space approach and the path integral formulation. Quantum mechanics in phase space (QMPS) thereby gives a third point of view which provides still more insight and understanding.

What does it get you?

[Quantum mechanics in phase space] can obviously shed light on subtle quantization problems as the comparison with classical theories is more systematic and natural. Since the variables involved are the same in both classical and quantum cases, the connection to the classical limit as ħ → 0 is more readily apparent. But beyond this and self-evident pedagogical intuition, what is this alternate formulation of QM and its panoply of satisfying mathematical structures good for?

It is the natural language to describe quantum transport, and to monitor decoherence of macroscopic quantum states in interaction with the environment, a pressing central concern of quantum computing. It can also serve to analyze and quantize physics phenomena unfolding in an hypothesized noncommutative spacetime with various noncommutative geometries. Such phenomena are most naturally described in Groenewold’s and Moyal’s language.

However, it may be fair to say that, as was true for the path integral formulation during the first few decades of its existence, the best QMPS “killer apps” are yet to come.

Dirac was not a fanI suspect the authors are not being quite fair to Dirac here. From the information I have, Dirac may simply have not been interested in a phase-space representation without the crucial probability interpretation. In a trivial sense, it’s obvious you can represent the quantum state in a “phase space” if that representation is allowed to be arbitrarily ugly.a :

A representative, indeed authoritative, opinion, dismissing even the suggestion that quantum mechanics can be expressed in terms of classical-valued phase space variables, was expressed by Paul Dirac in a letter to Joe Moyal on 20 April 1945…Dirac said, “I think it is obvious that there cannot be any distribution function F(p, q) which would give correctly the mean value of any f(p,q) …” He then tried to carefully explain why he thought as he did, by discussing the underpinnings of the uncertainty relation.

On the trials of Hip Groenewold:

Ever since his return from England in 1935 until his permanent appointment at theoretical physics in Groningen in 1951, Groenewold experienced difficulties finding a paid job in physics. He was an assistant to Zernike in Groningen for a few years, then he went to the Kamerlingh Onnes Laboratory in Leiden, and taught at a grammar school in the Hague from 1940 to 1942. There, he met the woman whom he married in 1942. He spent the remaining war years at several locations in the north of the Netherlands. In July 1945, he began work for another two years as an assistant to Zernike. Finally, he worked for four years at the KNMI (Royal Dutch Meteorological Institute) in De Bilt.

During all these years, Groenewold never lost sight of his research

The authors later give a long list of results where the phase-space formulation led to key insights. I can’t evaluate them, but I am familiar with Diosi and Kiefer’s elegant results on the finite-time positivity of phase-space representations, which has broadimplications for the fragility of entanglement under even weak decoherence.

Footnotes

(↵ returns to text)

I suspect the authors are not being quite fair to Dirac here. From the information I have, Dirac may simply have not been interested in a phase-space representation without the crucial probability interpretation. In a trivial sense, it’s obvious you can represent the quantum state in a “phase space” if that representation is allowed to be arbitrarily ugly.↵

One Comment

On the other side, instead of comparing CM and QM both in phase space formalisms, one can compare CM and QM both in Hilbert space formalisms, with a Hilbert space formalism for CM following Koopman’s lead from 1931. One could see my “An algebraic approach to Koopman classical mechanics”, https://arxiv.org/abs/1901.00526 (DOI in Annals of Physics 2020 there). Because there is the obvious parallel with phase space approaches that I point out above, I cite a recent review, J. Weinbub, D.K. Ferry, Appl. Phys. Rev. 5 (2018) 041104, http://dx.doi.org/10.1063/1.5046663.
There is also a large literature on the use of the Wigner function and other time-frequency functions in signal analysis, as in L. Cohen, Proc. IEEE 77 (1989) 941, http://dx.doi.org/10.1109/5.30749, which, however, is different in nature insofar as it is not probabilistic.